- #1
unscientific
- 1,734
- 13
Homework Statement
Part (a): Derive Ehrenfest's Theorem. What is a good quantum number?
Part (b): Write down the energy eigenvalues and sketch energy diagram showing first 6 levels.
Part (c): What's the symmetry of the new system and what happens to energy levels? Find a new good quantum number and corresponding operator. Use Ehrenfest's to show it is true.
Part (d): Write down the new wavefunctions.
Homework Equations
The Attempt at a Solution
Part (a)
[tex]\frac{d}{dt}<\psi|Q|\psi> = -<\psi|HQ|\psi> + i\hbar<\psi|\frac{dQ}{dt}|\psi> + <\psi|QH|\psi>[/tex]
[tex] = <\psi|QH - HQ|\psi> + i\hbar<\psi|\frac{dQ}{dt}|\psi>[/tex]
Assuming observable doesn't change with time:
[tex] = <\psi|[Q,H]|\psi>[/tex]
If ##[Q,H] = 0##, then Q and H share a common ket ##|\psi>##such that ##Q|\psi> = q_o|\psi>##, where ##q_0## is the good quantum number.
Stationary state is when ##\frac{d<Q>}{dt} = 0##.
Part (b)
[tex] E = (n_x + \frac{1}{2})\hbar \omega_x + (n_y + \frac{1}{2})\hbar \omega_y [/tex]
For ##\omega_x = \omega_y + \delta \omega##, ##E = \hbar \omega_y(n_x + n_y +1) + (n_x + \frac{1}{2})\hbar \delta \omega##.
Can I assume that energy contribution of ##\delta \omega## is small that when drawing the graph I can just ignore it?
In ascending order, the energy levels are: ##\hbar\omega_y##, ##2\hbar\omega_y##, ##3\hbar \omega_y## ... where (nx,ny): the first is nx = ny = 0, second is (0,1) or (1,0), the third is (1,1) or (2,0) or (0,2).
And how do I sketch the energy levels? Is it simply:
Part (c)
There's hardly any change to the energy diagram, since we ignored the contribution due to ##\delta \omega##?
I'm guessing the symmetry is the good quantum number ##n = n_x + n_y## is conserved? What do they mean by the corresponding differential operator?
Part (d)
I have no idea why the energy eigenstates will change. Are they the same as eigenfunctions?